Comment on the Scaling Function in AdS4 x CP3
aa r X i v : . [ h e p - t h ] J u l LPTENS 08/NNSPhT-t08/NNNITEP-TH-35/08
Comment on the Scaling Function in
AdS × CP Nikolay Gromov α , Victor Mikhaylov βα Service de Physique Th´eorique, CNRS-URA 2306 C.E.A.-Saclay, F-91191 Gif-sur-Yvette,France; Laboratoire de Physique Th´eorique de l’Ecole Normale Sup´erieure et l’Universit´eParis-VI, Paris, 75231, France; St.Petersburg INP, Gatchina, 188 300, St.Petersburg, Russia [email protected] β Institute for Theoretical and Experimental Physics,B. Cheremushkinskaya 25, Moscow 117259, Russia; Moscow Institute of Physics andTechnology, Institutsky per. 9, 141 700, Dolgoprudny, Russia [email protected]
Abstract
The folded spinning string in
AdS gives us an important insight into AdS/CF T duality.Recently its one-loop energy was analyzed in the context of
AdS /CF T by McLoughlin andRoiban arXiv:0807.3965, by Alday, Arutyunov and Bykov arXiv:0807.4400 and by KrishnanarXiv:0807.4561. They computed the spectrum of the fluctuations around the classical solution.In this paper we reproduce their results using the algebraic curve technique and show thatunder some natural resummation of the fluctuation energies the one-loop energy agrees perfectlywith the predictions of arXiv:0807.0777. This provides a further support of the all-loop Betheequations and of the AdS × CP algebraic curve developed in arXiv:0807.0437. Introduction
Integrability in
AdS/CF T duality [1] is an exciting subject in the modern theoretical physics.The integrability of 4D Yang-Mills theory was first discussed in [2, 3]. An intensive devel-opment started after the seminal paper by Minahan and Zarembo [4] where the long singletrace operators of N = 4 Super Yang-Mills theory, dual to the superstring in the AdS × S background, were mapped onto integrable spin chains. The string theory was shown to beclassically integrable [5] and is widely believed to be integrable at the quantum level as well.Many attempts were made towards complete understanding of the spectrum of this system.In particular, the asymptotic Bethe ansatz proposed in [6, 7, 8] was an important step in thisdirection.Recently, a new duality was discovered between N = 6 super Chern-Simons theory with U ( N ) × U ( N ) gauge group at level k and superstring theory in the AdS × CP background inthe large N limit with the ‘t Hooft coupling λ = N/k kept fixed [9, 10]. Amazingly, the gaugeside of the duality also exhibits integrability [11] (see also [12, 13]). The string theory turnsout to be classically integrable as well [14, 15]. The all-loop Bethe equations were proposedrecently in [16] and have already passed several independent tests [17, 18].An important consequence of the classical integrability is an existence of the finite gap al-gebraic curve [19, 20]. For the
AdS × CP superstring theory it was constructed in [21]. Thealgebraic curve describes all classical solutions in a unified gauge-invariant way and can be alsoused for computation of the spectrum of quantum fluctuations around a given classical solution[22]. We will use this technique to compute fluctuation frequencies around a particularly im-portant folded string classical solution. These frequencies were obtained recently in [23, 24, 25]by a direct diagonalization of the string action expanded around the classical solution. Thematch of our results provides a nontrivial test of the algebraic curve construction [21].The folded string is rotating in AdS with large spin S and with angular momentum J ∼ log S in CP . The difference between its energy and spin scales as J [26, 27] for all values ofthe ‘t Hooft coupling λ [9]. A similar solution in the context of AdS /CF T duality has beenalready well studied for all values of λ [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42,43, 44, 45, 46]. In [16] an equation describing the sl (2) sector of the AdS × CP string wasproposed: (cid:18) x + k x − k (cid:19) J = − S Y j = k x + k − x − j x − k − x + j ! − − / ( x + k x − j )1 − / ( x − k x + j ) σ ( u k , u j ) . (1.1)It describes, in particular, the folded string. One can see that the only difference with AdS × S string Bethe ansatz [47] is a minus sign on the r.h.s. It produces a simple redefinition of a scalingparameter ℓ AdS = J g log S → ℓ AdS = J h log S , (1.2)where g = √ λ Y M / π and h ( λ CS ) = p λ CS / O (1 / √ λ CS ) for large λ CS and h ( λ CS ) = λ CS + O ( λ CS ) for small λ CS [12, 48, 49, 50]. This implies, in turn, that once the energy of thestring in AdS × S is given by γ YM = J f ( g, ℓ AdS ) ℓ AdS , (1.3)2he energy of the folded string in AdS × CP reads [16] γ CS = J f ( h, ℓ AdS ) ℓ AdS , (1.4)where the one-loop scaling function f is given by [51] f ( h, ℓ ) ≃ (cid:16) √ ℓ + 1 − ℓ (cid:17) + √ ℓ + 1 − ℓ + 1) log (cid:0) ℓ (cid:1) − ( ℓ + 2) log √ ℓ +2 √ ℓ +1 − πh √ ℓ + 1 . (1.5)In the limit ℓ → f ( h, ≃ − πh . (1.6)Recently the folded string solution at one-loop was analyzed in AdS × CP [23, 24, 25] start-ing from a superstring action. In these papers the spectrum of fluctuations around the foldedstring was computed and summed up into a one-loop shift. A disagreement with Eqs.(1.4,1.5)was stated. According to these works, in the limit ℓ → − πh .In this paper we reproduce the results for the fluctuation energies from the finite gap alge-braic curve and propose a natural regularization of the sum of fluctuations leading precisely tothe conjectured result (1.4,1.5). In this section we recall the method of finding the fluctuation energies directly from the algebraiccurve [22]. The algebraic curve allows us to compute these fluctuation energies disregarding aparticular parametrization and gauge fixing of the superstring action. This provides us witha universal framework for computation of the one-loop corrections to the classical energies forsuperstrings living in different integrable backgrounds. The finite-gap solutions with any num-ber of cuts can be treated on equal grounds [52]. Moreover, the calculations are much simplerthen by the other methods. In particular, for the folded string solutions all the fluctuationscan be obtained just from comparing the algebraic curves for the
AdS × S and AdS × CP superstrings.We will start with describing the properties of the algebraic curves. The quasimomenta forthe classical solution of string sigma-model in AdS × CP are not independent, namely, { q , q , q , q , q } = −{ q , q , q , q , q } . (2.1)Here the quasimomenta q , q , q , q are responsible for the AdS part of sigma-model andthe others for the CP part [21].Since the motion of the folded string is constrained to AdS × S subspace, the AdS -quasimomenta q , q coincide with those for the folded string in AdS × S . They have twocuts shared with the functions q and q , respectively. The quasimomenta in CP are the sameas in the point-like string limit (BMN limit), because the motion in this subspace is trivial, q ( x ) = q ( x ) = − q ( x ) = − q ( x ) = 2 πνxx − ,q ( x ) = − q ( x ) = 0 . (2.2)3 Sfrag repla ementsq2q2q2 q3q3q3 q4q4q4 q5q5q5 q6q6q6 q7q7q7 q8q8q8 q9q9q9 +=(2; 9) (2; 5) (2; 6)
Figure 1:
Equation relating fermionic and bosonic fluctuations (2,5) and (2,9). Adding a pole withdouble residue between sheets 2 and 9 is equivalent to adding a (2,5) pole plus a (2,6) pole. Thefermionic fluctuations with polarizations (2,5) and (2,6) are equivalent since q = q . Let us briefly recall the idea of quantization of the string using the algebraic curve. Inorder to obtain quasiclassical frequencies one has to perturb the classical curve and find thecorresponding shift of the energy. These perturbations of the quasimomenta are implementedby adding some poles playing a role of infinitesimal cuts. We denote the polarization of theexcitation by a pair of numbers (i,j) corresponding to the sheets between which the pole is added.If one sheet corresponds to
AdS and another to CP , the fluctuation is fermionic, otherwise itis bosonic.The residues of the poles are fixed by quasiclassical quantization condition and their posi-tions are determined by the equations q i ( x ( i,j ) n ) − q j ( x ( i,j ) n ) = 2 πn ( i,j ) . (2.3)Thus the calculation splits into two steps: (i) calculating the response of the energy δE = Ω ij ( x )to insertion of a pole at some point x and (ii) solving the equations 2.3. Then the fluctuationenergy is simply δE ( i,j ) n = Ω ij ( x ( i,j ) n ).In our case one can notice that all the functions q i ( x ) have already appeared in the AdS × S algebraic curve, except for the trivial functions q and q . This allows us to write immediatelymost of the fluctuation frequencies. For example, the fermionic excitation energy δE (2 , n isobtained by adding a pole between a sheet with two cuts in the physical region and a BMN-likesheet (i.e. containing only two poles at ± AdS × S such fluctuation has polarization(ˆ2 , ˜3) and hence its frequency is equal to ω Fn given in Tab.1. In the same way one finds theother δE ( i,j ) n ’s, not involving the sheets 5 and 6, which we call “heavy” fluctuations.For the fluctuations δE ( i,j ) n with j = 5 or 6, to which we refer as “light”, one can also avoidexplicit calculations using a simple trick illustrated in Fig.1 and Fig.2. Let us for exampleconsider Fig.1. The prescription of [21] says that for the (2,9) and (1,10) fluctuations theresidue should be doubled compared to other excitations. By that reason we depict them bya double line. This configuration can be thought of as adding a (2,5) pole (and automaticallya “mirror” (6,9) pole, due to (2.1)), plus a pole between the second and the sixth sheets (and4 Sfrag repla ementsq2q2q2 q3q3q3 q4q4q4 q5q5q5 q6q6q6 q7q7q7 q8q8q8 q9q9q9 +=(3; 7) (3; 5) (4; 6)
Figure 2:
Equation relating fluctuations inside CP with polarizations (3,7) and (3,5). The (3,7)fluctuation can be decomposed into a (3,5) fluctuation plus (4,7) fluctuation. The fluctuations (3,5)and (4,7) for the folded string are obviously equivalent. hence also a (5,9) pole). The poles on the intermediate sheets have opposite residues andcancel. Both configurations on the right hand side of Fig.1 are equivalent and correspond tothe fermionic fluctuation with polarization (2,5). Hence we find that Ω ( x ) = Ω ( x ).At the next stage one has to solve the equation for the positions of the poles. From (2.1)and (2.2) it is clear that the solutions for the positions of the polarizations (2,5) and (2,9) arerelated by x n = x n . (2.4)Collecting all together we conclude that ω (2 , n = ω (2 , n . The similar trick gives the remainingfrequencies ω (1 , n and ω (3 , n as one can see from the Fig.2.Table 1: Notations for the frequencies of the folded string for J ∼ log S → ∞ . The frequenciesare taken from [51]. In the table we use notations ν = J πh , νκ = ℓ √ ℓ +1 . eigenmodes notation p n + 2 κ ± √ κ + n ν √ n + 2 κ − ν ω A ± n ω An √ n + κ ω Fn √ n + ν ω Sn frequency multiplicity polarizationsAdS ω A + n ω A − n ω An × × × , , , fermions ω Fn ω A + n / ω A − n / × × × , , , , , , , , CP ω Sn ω S n / × × , , , , , Having the fluctuation frequencies ω ( i,j ) n computed we are ready to compute the shift of theclassical value of the energy due to zero point oscillations. Formally one can write δE − loop = 12 X ( i,j ) ( − F ( i,j ) X n ω ( i,j ) n , (3.1)however each sum over n for a given polarization ( i, j ) is divergent. A naive regularizationwhich indeed leads to a finite result is to interchange the order of summation over n and ( i, j ).Then the cancelations between fermions and bosons make the sum over n convergent.In the context of AdS × S this regularization leads to the agreement with the Bethe ansatzprediction. However, repeating the same procedure in AdS × CP leads to a disagreement[23, 24, 25] with the prediction from the all-loop Bethe ansatz [16]. In this section we willargue that this regularization is not natural for the case of AdS × CP . We present anotherregularization which leads to a different finite result. We argue by different means that ourregularization is indeed the one which should be used.The main difference between AdS × CP and AdS × S theories is existence of two dispersionrelations in the BMN spectrum. Accordingly the BMN spectrum [12, 48, 49] of AdS × CP is naturally divided into two groups. Because of the differences in the diameters of AdS and CP there are four fermions, three AdS fluctuations and one CP fluctuation which have theenergies E n = 1 κ √ n + κ , (3.2)whereas the other four CP and four fermionic BMN fluctuations have the following spectrum ǫ n = 12 κ √ n + κ . (3.3)6n the Bethe ansatz language the first group of the fluctuations has both momentum carryingroots u and u ¯4 excited, while the second group has only one of them. We call them heavy andlight excitations respectively. For even n one can think about the fluctuations from the firstgroup to be some kind of bound state of two “light” excitations (with zero binding energy): E n = ǫ n/ + ǫ n/ . (3.4)Indeed, from this argument we see that for even n “heavy” fluctuations with mode number n and the “light” fluctuations with mode number n/ . We define K n = (cid:26) ω heavy n + ω light n/ n ∈ even ω heavy n n ∈ odd (3.5)where ω heavy n and ω light n in general are defined by ω heavy n = ω (1 , n + ω (2 , n + ω (1 , n − ω (1 , n − ω (1 , n − ω (2 , n − ω (2 , n + ω (3 , n (3.6) ω light n = ω (3 , n + ω (3 , n + ω (4 , n + ω (4 , n − ω (1 , n − ω (1 , n − ω (2 , n − ω (2 , n . We claim that the one-loop shift of the string energy is given by E − loop = lim N →∞ N X n = − N K n κ . (3.7)Using the notations given in Tab.2 one can pass to the fluctuations listed in Tab.1. For example ω (1 , n = ω A + n and ω (3 , n = ω S n /
2. From Eq.(3.6) we get ω heavy n = ω A + n + ω A − n + ω An − ω Fn + ω Sn (3.8) ω light n/ = 2 ω Sn − ω A + n − ω A − n . For large κ we can replace the sum with an integral E − loop ≃ lim N →∞ Z N − N ω heavy n + ω light n κ dn = Z ∞ (cid:0) ω An + ω A + n + ω A − n + 4 ω Sn − ω Fn (cid:1) dn κ . (3.9)We notice that exactly this integral was computed in [51]. We immediately write the result E − loop ≃ Jℓ √ ℓ + 1 − ℓ + 1) log (cid:0) ℓ (cid:1) − ( ℓ + 2) log √ ℓ +2 √ ℓ +1 − πh √ ℓ + 1 (3.10)where we used that νκ = ℓ √ ℓ +1 and ν = J πh . Eq.(3.10) agrees completely with Eqs.(1.4,1.5)! Inparticular, when ℓ → E − loop ≃ − π log S . (3.11) Formally one can pass to the BMN limit ν = κ to distinguish them. Summary
In this paper we propose a particular regularization of the sum over the fluctuation frequencies.The regularization goes as follows: we split the fluctuations into two groups with 4 bosons and4 fermions each. We call them heavy and light fluctuations since for even mode numbers n heavy ones can be decomposed into a sum of two light fluctuations with mode number n/
2. Thelight fluctuations are in some sense more fundamental. We prescribe then the mode number2 n heavy fluctuations to be treated together with mode number n light fluctuations. Followingthis prescription we match the one-loop energy shift with the prediction from the all-loopBethe equations. This regularization method looks a bit artificial on the n plane, but from thealgebraic curve point of view it makes perfect sense. The positions of the excitations on thealgebraic curve of the light and heavy excitations are related in the following way x heavy2 n = x light n (4.1)(see for example Eq.(2.4)). We can regularize the sum over the fluctuations in terms of thealgebraic curve by introducing an ǫ balls around x = ± AdS × S string this ǫ regularization leads to thesimple cut-off prescription in the sum over fluctuations. However, in the AdS × CP case oneshould be more careful.Whereas it is clear from the algebraic curve why this particular regularization makes senseit is rather hard to justify it starting from the world-sheet string action. At the end of the daythe algebraic curve and the action are two equivalent descriptions of the semiclassical stringsand it should be possible to understand this regularization from both points of view. One of thepossible approaches to better understanding of the problem is to pass to the Frolov-Tseytlinlimit ( J → ∞ and J/h large) where the conjectured all-loop Bethe equations reduce to thetwo-loop Bethe equations derived from the CS perturbation theory [11]. Our hope is that thefinite-size corrections to the scaling limit of the two-loop Bethe equations will only be consistentwith the sum over fluctuations if our prescription is used. Acknowledgments
We would like to thank L. Alday, G. Arutyunov, C. Krishnan, I. Kostov, T. McLoughlin,R. Roiban, S. Schafer-Nameki, D. Serban, I. Shenderovich, A. Tirziu, A. Tseytlin, D. Volin,K. Zarembo and especially P. Vieira for many useful discussions. V.M. would also like tothank A. Gorsky for introducing into the topic of integrability in AdS/CFT. NG was partiallysupported by RSGSS-1124.2003.2, by RFBR grant 08-02-00287 and ANR grant INT-AdS/CFT(contract ANR36ADSCSTZ). The work of V.M. was partially supported by RFFI grant 06-02-17382, grant for support of scientific schools NSh-3036.2008.2 and CNRS-RFBR grant PICS-07-0292165. This work was done during our stay at Les Houches Summer School, which wethank for hospitality. 8 ibliography [1] J. M. Maldacena, “The large N limit of superconformal field theories and supergravity,”
Adv. Theor. Math. Phys. (1998) 231–252, arXiv:hep-th/9711200 .[2] L. N. Lipatov, “High-energy asymptotics of multicolor QCD and exactly solvable latticemodels,” arXiv:hep-th/9311037 .[3] L. D. Faddeev and G. P. Korchemsky, “High-energy QCD as a completely integrablemodel,” Phys. Lett.
B342 (1995) 311–322, arXiv:hep-th/9404173 .[4] J. A. Minahan and K. Zarembo, “The Bethe-ansatz for N = 4 super Yang-Mills,”
JHEP (2003) 013, arXiv:hep-th/0212208 .[5] I. Bena, J. Polchinski, and R. Roiban, “Hidden symmetries of the AdS(5) x S**5superstring,” Phys. Rev.
D69 (2004) 046002, arXiv:hep-th/0305116 .[6] M. Staudacher, “The factorized S-matrix of CFT/AdS,”
JHEP (2005) 054, hep-th/0412188 .[7] N. Beisert and M. Staudacher, “Long-range psu (2 , |
4) Bethe ansaetze for gauge theoryand strings,”
Nucl. Phys.
B727 (2005) 1–62, hep-th/0504190 .[8] N. Beisert, B. Eden, and M. Staudacher, “Transcendentality and crossing,”
J. Stat.Mech. (2007) P021, hep-th/0610251 .[9] O. Aharony, O. Bergman, D. L. Jafferis, and J. Maldacena, “N=6 superconformalChern-Simons-matter theories, M2-branes and their gravity duals,” arXiv:0806.1218 [hep-th] .[10] M. Benna, I. Klebanov, T. Klose, and M. Smedback, “Superconformal Chern-SimonsTheories and AdS4/CFT3 Correspondence,” arXiv:0806.1519 [hep-th] .[11] J. A. Minahan and K. Zarembo, “The Bethe ansatz for superconformal Chern-Simons,” arXiv:0806.3951 [hep-th] .[12] D. Gaiotto, S. Giombi, and X. Yin, “Spin Chains in N=6 SuperconformalChern-Simons-Matter Theory,” arXiv:0806.4589 [hep-th] .[13] D. Bak and S.-J. Rey, “Integrable Spin Chain in Superconformal Chern-Simons Theory,” arXiv:0807.2063 [hep-th] .[14] G. Arutyunov and S. Frolov, “Superstrings on AdS4 x CP3 as a Coset Sigma-model,” arXiv:0806.4940 [hep-th] .[15] j. Stefanski, B., “Green-Schwarz action for Type IIA strings on
AdS × CP ,” arXiv:0806.4948 [hep-th] .[16] N. Gromov and P. Vieira, “The all loop AdS4/CFT3 Bethe ansatz,” arXiv:0807.0777 [hep-th] . 917] D. Astolfi, V. G. M. Puletti, G. Grignani, T. Harmark, and M. Orselli, “Finite-sizecorrections in the SU(2) x SU(2) sector of type IIA string theory on AdS4 x CP3,” arXiv:0807.1527 [hep-th] .[18] C. Ahn and R. I. Nepomechie, “N=6 super Chern-Simons theory S-matrix and all-loopBethe ansatz equations,” arXiv:0807.1924 [hep-th] .[19] V. A. Kazakov, A. Marshakov, J. A. Minahan, and K. Zarembo, “Classical / quantumintegrability in AdS/CFT,” JHEP (2004) 024, hep-th/0402207 .[20] N. Beisert, V. A. Kazakov, K. Sakai, and K. Zarembo, “The algebraic curve of classicalsuperstrings on AdS(5) x S(5),” Commun. Math. Phys. (2006) 659–710, hep-th/0502226 .[21] N. Gromov and P. Vieira, “The AdS4/CFT3 algebraic curve,” arXiv:0807.0437 [hep-th] .[22] N. Gromov and P. Vieira, “The AdS(5) x S(5) superstring quantum spectrum from thealgebraic curve,”
Nucl. Phys.
B789 (2008) 175–208, hep-th/0703191 .[23] T. McLoughlin and R. Roiban, “Spinning strings at one-loop in AdS4 x P3,” arXiv:0807.3965 [hep-th] .[24] L. F. Alday, G. Arutyunov, and D. Bykov, “Semiclassical Quantization of SpinningStrings in AdS4 x CP3,” arXiv:0807.4400 [hep-th] .[25] C. Krishnan, “AdS4/CFT3 at One Loop,” arXiv:0807.4561 [hep-th] .[26] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, “A semi-classical limit of thegauge/string correspondence,”
Nucl. Phys.
B636 (2002) 99–114, arXiv:hep-th/0204051 .[27] A. V. Belitsky, A. S. Gorsky, and G. P. Korchemsky, “Logarithmic scaling in gauge /string correspondence,”
Nucl. Phys.
B748 (2006) 24–59, arXiv:hep-th/0601112 .[28] A. V. Kotikov and L. N. Lipatov, “On the highest transcendentality in N = 4 SUSY,”
Nucl. Phys.
B769 (2007) 217–255, arXiv:hep-th/0611204 .[29] L. F. Alday, G. Arutyunov, M. K. Benna, B. Eden, and I. R. Klebanov, “On the strongcoupling scaling dimension of high spin operators,”
JHEP (2007) 082, arXiv:hep-th/0702028 .[30] I. Kostov, D. Serban, and D. Volin, “Strong coupling limit of Bethe ansatz equations,” Nucl. Phys.
B789 (2008) 413–451, arXiv:hep-th/0703031 .[31] M. Beccaria, G. F. De Angelis, and V. Forini, “The scaling function at strong couplingfrom the quantum string Bethe equations,”
JHEP (2007) 066, arXiv:hep-th/0703131 . 1032] I. Kostov, D. Serban, and D. Volin, “Functional BES equation,” arXiv:0801.2542 [hep-th] .[33] P. Y. Casteill and C. Kristjansen, “The Strong Coupling Limit of the Scaling Functionfrom the Quantum String Bethe Ansatz,” Nucl. Phys.
B785 (2007) 1–18, arXiv:0705.0890 [hep-th] .[34] A. V. Belitsky, “Strong coupling expansion of Baxter equation in N=4 SYM,”
Phys. Lett.
B659 (2008) 732–740, arXiv:0710.2294 [hep-th] .[35] R. Roiban, A. Tirziu, and A. A. Tseytlin, “Two-loop world-sheet corrections in AdS5 xS5 superstring,”
JHEP (2007) 056, arXiv:0704.3638 [hep-th] .[36] L. F. Alday and J. M. Maldacena, “Comments on operators with large spin,” JHEP (2007) 019, arXiv:0708.0672 [hep-th] .[37] B. Basso, G. P. Korchemsky, and J. Kotanski, “Cusp anomalous dimension in maximallysupersymmetric Yang- Mills theory at strong coupling,” Phys. Rev. Lett. (2008) 091601, arXiv:0708.3933 [hep-th] .[38] R. Roiban and A. A. Tseytlin, “Strong-coupling expansion of cusp anomaly fromquantum superstring,”
JHEP (2007) 016, arXiv:0709.0681 [hep-th] .[39] R. Roiban and A. A. Tseytlin, “Spinning superstrings at two loops: strong-couplingcorrections to dimensions of large-twist SYM operators,” Phys. Rev.
D77 (2008) 066006, arXiv:0712.2479 [hep-th] .[40] D. Fioravanti, P. Grinza, and M. Rossi, “Strong coupling for planar N = 4 SYM theory:an all-order result,” arXiv:0804.2893 [hep-th] .[41] L. Freyhult, A. Rej, and M. Staudacher, “A Generalized Scaling Function forAdS/CFT,” arXiv:0712.2743 [hep-th] .[42] B. Basso and G. P. Korchemsky, “Embedding nonlinear O(6) sigma model into N=4super-Yang- Mills theory,” arXiv:0805.4194 [hep-th] .[43] D. Fioravanti, P. Grinza, and M. Rossi, “The generalised scaling function: a note,” arXiv:0805.4407 [hep-th] .[44] F. Buccheri and D. Fioravanti, “The integrable O(6) model and the correspondence:checks and predictions,” arXiv:0805.4410 [hep-th] .[45] N. Gromov, “Generalized Scaling Function at Strong Coupling,” arXiv:0805.4615 [hep-th] .[46] M. Beccaria, “The generalized scaling function of AdS/CFT and semiclassical stringtheory,” arXiv:0806.3704 [hep-th] . 1147] G. Arutyunov, S. Frolov, and M. Staudacher, “Bethe ansatz for quantum strings,” JHEP (2004) 016, hep-th/0406256 .[48] G. Grignani, T. Harmark, and M. Orselli, “The SU(2) x SU(2) sector in the string dualof N=6 superconformal Chern-Simons theory,” arXiv:0806.4959 [hep-th] .[49] T. Nishioka and T. Takayanagi, “On Type IIA Penrose Limit and N=6 Chern-SimonsTheories,” arXiv:0806.3391 [hep-th] .[50] I. Shenderovich, “Giant magnons in AdS /CF T : dispersion, quantization and finite–sizecorrections,” arXiv:0807.2861 [hep-th] .[51] S. Frolov, A. Tirziu, and A. A. Tseytlin, “Logarithmic corrections to higher twist scalingat strong coupling from AdS/CFT,” Nucl. Phys.
B766 (2007) 232–245, arXiv:hep-th/0611269arXiv:hep-th/0611269